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1 Answer
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Nocturnal lagophthalmos is the search term you're looking for. But it's less that they "can" fall asleep with their eyes open and more that they "can't" close their eyes during sleep:

Nocturnal lagophthalmos is the inability to close the eyelids during
sleep. Lagophthalmos is associated with exposure keratopathy, poor
sleep, and persistent exposure-related symptoms. There are a variety
of causes of lagophthalmos, grouped as proptosis/eye exposure
etiologies and palpebral insufficiency etiologies.

My step-father opens his eyes during his sleep. He is actually asleep and he does relate a positive experience from sleep. I want to add to Keegan's answer that Nocturnal lagophthalmos can cause vision problems like keratopathy. medical-dictionary.thefreedictionary.com/acid-fast+keratopathy
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user3832Feb 9 '14 at 15:30